Abstract

To prove Kronecker’s density theorem in Bishop-style constructive analysis one needs to define an
irrational number as a real number that is bounded away from each rational number. In fact, once one
understands “irrational” merely as “not rational”, then the theorem becomes equivalent
to Markov’s principle. To see this we undertake a systematic classification, in the vein of constructive
reverse mathematics, of logical combinations of “rational” and “irrational” as predicates
of real numbers.